Deformation Of Surfaces In Sphere, Affine And Similarity Geometries

The motions of curves and surfaces have a wide range of applications. It is well known that several nonlinear phenomena in mathematics,physics, chemistry,biology and so on,modeled by nonlinear differential equations,can describe also the evolution of surfaces in time.So many nonlinear evolution equations have been shown to be closely related to motions of curves and surfaces.In particular,many integrable equations arise naturally from motions of curves and surfaces.In this paper,we study deformation of surfaces induced by endowing with one or two extra space variables to the formulae of the curve motions in S~2(R), S~3(R),affine and higher-dimensional similarity geometries,respectively.In this way,the relationship between deformation of surfaces and integrable equations are set up.In Chapter 1,we recall the history of the motions of curves and surfaces in briefly.Deformation of surfaces in S~2(R),S~3(R)and affine geometries are studied in Chapter 2,which is derived by endowing with an extra space variable y to the formulae of the curve motions in these geometries,and induces a 2+1-dimensional generalization to the sine-Gordon equation,a 2+1-dimensional complex mKdV equation,a 2+1-dimensional breaking soliton equation and a 2+1-dimensional nonlinear evolution equation to be regarded as a generalization to the 1+1-dimensional KdV equation,respectively.In Chapter 3,we discuss surface motions in higher-dimensional similarity geometries,which are derived by endowing with one or two extra space variables to the formulae of the curve motions in higher-dimensional similarity geometries.It is to be shown that higher-dimensional integrable systems arise from such motions.